At the present state-of-the-art, the simplest inflationary models, based either on scalar fields in General Relativity or on modified f(R) gravity, which produce the best fit to all existing astronomical data require one, maximum two dimensionless parameters taken from observations only. The main discoveries expected for these models in future are discussed. Among them the most fundamental are primordial quantum gravitational waves generated during inflation. In one parametric models, including the original R+R^2 one, the definite prediction for the tensor-to-scalar ratio r=3(1-n_s)^2=0.004 follows. The role of one-loop quantum gravitational corrections to these models is considered. Inflation, as a metastable quantum state, had finite life-time, and differences in its duration in terms of the number of e-folds between various points of space can be determined with remarkable accuracy. In the models considered, the most generic predecessor of inflation is an anisotropic and inhomogeneous space-time near a generic space-like singularity. Since the transition from such space-time to the generalized quasi-de Sitter regime is generic, too, for inflation to begin inside a patch including the observable part of the Universe, causal connection inside the whole patch is not necessary. However, it becomes obligatory for a graceful exit from inflation in order to have practically the same number of e-folds during inflation inside this patch.